Multigraph Conditions for Multistability, Oscillations and Pattern Formation in Biochemical Reaction Networks

We represent interactions among biochemical species using a directed multigraph, which is a generalization of a more commonly used digraph. We show that network properties that are known to lead to multistability or oscillations, such as the existence of a positive feedback cycle, can be generalized to ldquocritical subnetworksrdquo that can contain several cycles. We also derive corresponding graph-theoretic conditions for pattern formation for the respective reaction-diffusion models. We present as an example a model for cell cycle and apoptosis along with bifurcation diagrams and sample solutions that confirm the predictions obtained with the help of the multigraph network conditions.